The Lang-Trotter conjecture for Drinfeld modules by David Tweedle A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Pure Mathematics Waterloo, Ontario, Canada, 2011 c David Tweedle 2011 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Abstract In 1986, Gupta and Murty proved the Lang-Trotter conjecture in the case of elliptic curves having complex multiplication, conditional on the generalized Riemann hypothesis. That is, given a non-torsion point P 2 E(Q), they showed that P (mod p) generates E(Fp) for infinitely many primes p, conditional on the generalized Riemann hypothesis. We demonstrate that Gupta and Murty's result can be translated into an unconditional result in the language of Drinfeld modules. We follow the example of Hsu and Yu, who proved Artin's conjecture unconditionally in the case of sign normalized rank one Drinfeld modules. Further, we will cover all necessary background information. v Acknowledgements I would like to thank my lovely wife Kemba for her love, support and encouragement. I would like to thank my parents Terry and Bonnie Tweedle for all their help over the years. I would like to thank my supervisor, Professor Wentang Kuo, for his patience and guidance. I would like to thank Professors Yu-Ru Liu, Kevin Hare, David McKinnon for being helpful and wise. I would like to thank Jesse for being a good brother and friend. My thanks go to Professors Kumar Murty, David Jao and my entire committee for all their hard work and helpful comments on my thesis, your comments are greatly appreciated. Thank you to Mrs. Yvonne Thomas for welcoming me into your family and for your unending support. My thanks go to Mr. Enos Thomas for sharing his advice and experience with me. My thanks also to my aunts Karen and Donna, my uncle Frank, my cousins Kim, Kelly, Mark and Scott and my grandma Shirley. vii Dedication To Kemba! ix Table of Contents Nomenclature xiii 1 Introduction 1 1.1 Artin's primitive root conjecture . 1 1.2 The Lang-Trotter Conjecture . 3 1.3 Gupta's and Murty's work . 4 1.4 The Carlitz module and Drinfeld modules . 5 1.5 Artin's conjecture for Drinfeld modules . 7 1.6 Lang-Trotter for rank 2 Drinfeld modules . 8 1.7 An overview of the rest of the thesis . 11 2 Overview of the Lang-Trotter conjecture and Gupta's and Murty's work 13 2.1 Elliptic Curves review . 13 2.2 Main Results . 15 2.3 The Lang-Trotter condition . 17 2.4 Algebra, Kummer and Discriminants . 19 2.5 Analysis overview of CM case . 20 2.6 Free subgroups of high rank . 22 2.7 Density calculation . 25 xi 3 Function Fields and Drinfeld Modules 27 3.1 Number theory in function fields . 27 3.2 Additive polynomials . 30 3.3 Valued fields and Newton polygons . 34 3.4 Drinfeld modules . 36 3.5 Fundamental structures for Drinfeld modules . 37 3.6 Complex Multiplication . 38 3.7 Reduction of Drinfeld modules . 39 3.8 Analytic theory . 41 3.9 Class field theory . 43 3.10 Drinfeld modules over global function fields . 45 4 Number Theory 47 4.1 Lang-Trotter conditions . 47 4.2 Basic discriminant overview . 53 4.3 All discriminant results . 54 4.4 Kummer theory of Hsu and Yu . 62 4.5 Kummer theory of Ribet and Bashmakov . 70 4.6 Kummer theory for Drinfeld modules . 71 5 Analysis and proof of main results 75 5.1 Main Results . 75 5.2 Effective Chebotarev Density Theorem . 77 5.3 Main term of Theorem 5.1.1 ......................... 78 5.4 Proof of Theorem 5.1.2 ............................ 87 6 Future work 95 References 99 xii Nomenclature δa; δE(a); δφ(a); δE(Γ); δφ(Γ) Densities of primes for Artin-like problems Γ A finitely generated torsion-free submodule of either E(Q) or φ(F ) 1 A fixed rational prime of F Fr The finite field with r elements τ The r-power Frobenius map, τ(X) = Xr A The ring of elements of F which are regular everywhere except possibly 1 a A non-torsion point of E(Q) or φ(F ) E An elliptic curve defined over Q F A function field, i.e. an extension of Fr having transcendence degree 1 over Fr with F \ Fr = Fr F fτg The twisted polynomial ring F fτg k This refers to either a quadratic imaginary extension of Q or an extension of F where 1 does not split a 1=m −1 Km = Q(a ; ζm); Q(m a; E[m]) The particular extension we are talking about will be clear from context ~ Na(x);NΓ(x); NΓ(x) Prime counting functions for Artin-like problems P; P; p; q Finite primes in various function fields p; q Rational primes xiii Chapter 1 Introduction 1.1 Artin's primitive root conjecture We say that an integer a is a primitive root mod p if the class of a mod p generates the ∗ multiplicative group Fp, where p is prime. In [2], Artin conjectured that an integer a, which is not equal to ±1 or any square, should be a primitive root for infinitely many primes p. He further conjectured that the set of primes p for which a is a primitive root mod p should have a natural density A(a) > 0 which is given by an Euler product. These conjectures are called Artin's primitive root conjecture and have been a continuing subject of study to this date. We need algebraic number theory to investigate the density A(a). For each positive a 1=m integer m, let Km = Q(a ; ζm), where ζm is a primitive mth root of unity. By examining the behaviour of the prime ideal p in the extensions Kq for q prime, we get that a is a a primitive root mod p if and only if p does not split completely in any extension Kq , for q prime. The Chebotarev density theorem then tells us that the density of primes which a a −1 split completely in Kq is [Kq : Q] . (For an effective version see [19]) This heuristic allows us to guess that Y 1 A(a) = 1 − : [Ka : ] q prime q Q By incorporating inclusion-exclusion into our heuristic, we obtain the more accurate for- mula X µ(m) A(a) = ; [Ka : ] m≥1 m Q 1 where µ(m) = (−1)j if m is the product of j distinct primes for some j, and 0 otherwise. In 1967, Hooley[16] proved Artin's primitive root conjecture conditionally on the gener- a alized Riemann Hypothesis for the number fields Km; m ≥ 1. Recall that the generalized a Riemann Hypothesis is for Km is the assumption that the analytic continuation of the function X −s a a ζKm (s) = NKm=Q(I) ; I an ideal of O a Km has all of its non-trivial zeroes lying on the line Re(s) = 1=2. Now, let Na(x) = #fp prime : p ≤ x; a is a primitive root mod pg: Theorem 1.1.1 ([16], Theorem, pp. 219-220). Suppose that a is a non-square and not equal to ±1. Further, assume that the generalized Riemann Hypothesis holds for the exten- a sions Km; m ≥ 1. Then 2 Na(x) = A(a)x= log x + O(x log log x=(log x) ); with X µ(m) A(a) = > 0: [Ka : ] m≥1 m Q We can broadly think of Hooley's proof as being organized into several steps, which we can imitate when thinking of similar questions. First, restate \a is a primitive root mod a p" as \p splits completely in no field Km". Next, determine a bound for the discriminant a a of Km for each m ≥ 1, and determine bounds for [Km : Q] for each m ≥ 1. One then observes that Na(x) = N(x; y) + O(M(y; x)); a where N(x; y) is the number of primes which do not split completely in Kq with q ≤ y, a and M(y; x) is the number of primes which split completely in some Kq with y < q ≤ x. Then see that y can be chosen so that N(x; y) = A(a)x= log x + O(x log log x=(log x)2). Then the error term M(y; x) is dealt with by a combination of sieving techniques. Finally, a further investigation of [Km : Q] gives that A(a) has an Euler product, which leads to the fact that A(a) > 0. After Hooley's success, it is natural to ask whether the concept of a primitive root can be generalized to (abelian) algebraic groups defined over Q. One particular example is elliptic curves. 2 1.2 The Lang-Trotter Conjecture 2 2 3 S Suppose that the algebraic curve E(C) = f(x; y) 2 C : y = x + ax + bg f1g is smooth (this is the same as requiring that f(x) = x3 + ax + b has no repeated roots). Suppose also that a; b 2 Q. Then we say that E is an elliptic curve defined over Q. The set 2 2 3 S of rational points of E(Q) = f(x; y) 2 Q : y = x + ax + bg f1g forms an abelian group.